Does uniform convergence imply absolute convergence for Fourier series? Let $f : \mathbb R \to \mathbb R$ be a continuous $1$-periodic function. We can compute the Fourier coefficients
$$a_n := \int_0^1 f(x)\exp(-2\pi i nx) dx.$$
If
$$\sum_{n \in \mathbb Z} |a_n| < \infty $$
it's easy to see that
$$\sum_{n \in \mathbb Z} a_n\exp(2\pi i n x)$$
converges uniformly to $f$.
Is the opposite true as well? So does the uniform convergence of the last series imply the absolute convergence of the Fourier coefficients?
 A: From Edwards' Fourier Series: A Modern Introduction, Volume 1:
Theorem 7.2.2. (1) Suppose that $a_n\downarrow 0$ (meaning $a_n\ge 0$ is monotonically nonincreasing and $a_n\to 0$). Then the series $\sum_{n=1}^\infty a_n \sin(nx)$ is uniformly convergent, if and only if $na_n \to 0$.
In particular, we can say that $\sum_{n=2}^\infty \frac{\sin(nx)}{n\log n}$ converges uniformly on an interval of length $2\pi$, but not absolutely since $\sum_{n=2}^\infty \frac{1}{n\log n}$ diverges.
If you take the uniform boundedness of the series $\sum_{n=1}^\infty \frac{\sin nx}{n}$ on intervals of length $2\pi$ for granted, you can also give a direct proof that $\sum_{n=2}^\infty\frac{\sin(nx)}{n\log n}$ converges uniformly using the method of summation by parts.
Added:

Corollary to Theorem 7.16 in Rudin's PMA (paraphrased). Suppose $f_n$ are Riemann integrable on $[a,b]$, and that
$$f(x) = \sum_{n=1}^\infty f_n(x)\quad(a\le x\le b),$$
the series converging uniformly on $[a,b]$. Then $f$ is Riemann integrable on $[a,b]$, and
$$
\int_a^b f(x)\,dx = \sum_{n=1}^\infty \int_a^bf_n(x)\,dx.
$$
In other words, the series may be integrated term by term.

Note that we do not require absolute convergence of the series $\sum f_n(x)$ to integrate term by term.
Thus, for a sequence $a_n$ as in Edwards' Theorem 7.2.2. (1) and a function $f(x) = \sum a_n\sin(nx)$, the coefficients $a_n$ are the Fourier coefficients of $f(x)$:
$$
a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x)\sin(nx)\,dx.
$$

Also, the discussion in the answer and comments on this post is relevant, and in particular pointed me to the book of Edwards.
